Geraghty and Ćirić type fixed point theorems in b-metric spaces

Abstract

In this paper, we obtain some fixed point theorems for admissible mappings in b-metric spaces. Some useful examples are given to illustrate the facts. We also discuss an application to a nonlinear quadratic integral equation. Our results complement, extend and generalize a number of fixed point theorems including the well-known Geraghty [M. A. Geraghty, Proc. Amer. Math. Soc., 40 (1973), 604-608]

Some New Observations on Geraghty and Ćirić Type Results in b-Metric Spaces

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We discuss recent fixed point results in b-metric spaces given by Pant and Panicker (2016). Our results are with shorter proofs.

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Fixed point results for admissible mappings with application to integral equations

“…Throughout the whole proof of Theorem 1.18 (that is, Theorem 4.4 of [11], the proof of the uniqueness of fixed point is incorrect. This is because the authors from [11] increased some additional conditions such as α(x * , T x * ) ≥ 1, α(y * , T y * ) ≥ 1, β(x * , T x * ) ≥ 1, and β(y * , T y * ) ≥ 1 when T is continuous, whereas these conditions hold only if X is (α, β)-regular. However, these conditions did not appear in the hypotheses of Theorem 1.18.…”

“…[11] has some problems. Since α(x n , x n+1 ) ≥ 1 and β(x n , x n+1 ) ≥ 1 for all n ∈ N, then clearly, α(x n k , x n k +1 ) ≥ 1 and β(x n k , x n k +1 ) ≥ 1 for all k ∈ N. Hence both Theorem 4.4 of [11] and Theorem 2.1 of [4] made some mistakes.…”

“…There is now extensive variety of literature dealing with fixed point problems via α-admissible mappings (see [4,14,15,19,20]). The purpose of this paper is to do some work to improve and optimize the main results of [11] and [19]. Our conditions are much wider and more applicable.…”

SOME FIXED POINT THEOREMS FOR GENERALIZED $\alpha$-GERAGHTY CONTRACTION TYPE MAPPINGS IN $B$-METRIC~SPACES AND SOME APPLICATIONS TO THE NONLINEAR INTEGRAL EQUATION

“…The first important difference between a metric and a b-metric is that the b-metric need not be a continuous function in its two variables, see [22,Example 13]. This led to many fixed point theorems on b-metric spaces being stated, so the readers may refer to [1,3,6,8,17,18,19,20,24,25,28,29] and references therein.…”